Problem 3 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$3 x+4=19 ; 5$$

Step-by-Step Solution

Answer

Yes, \(x = 5\) is the solution to the equation.

Step 1: Identify the Variable and Substitute

First, identify the variable in the equation, which is \(x\). Then, substitute \(x = 5\) into the equation \(3x + 4 = 19\). This involves replacing every occurrence of \(x\) with 5.

Step 2: Perform the Multiplication

Substitute \(x = 5\) into the equation and perform the multiplication: \(3 \times 5 + 4\). Calculate \(3 \times 5 = 15\).

Step 3: Add the Constant Term

Add the result of the multiplication to 4: \(15 + 4 = 19\).

Step 4: Compare the Calculated Result to 19

The left side of the equation simplifies to 19. Since 19 equals the right side of the equation, \(x = 5\) is indeed the solution to the equation.

Key Concepts

Variable Identification

In the realm of linear equations, identifying the variable is a vital first step in solving them. A variable is essentially a symbol, often a letter like \(x\), that represents an unknown number. In the equation \(3x + 4 = 19\), the variable is \(x\). This symbol holds a place for a value that we need to determine. A quick tip to remember: variables are placeholders for numbers we aim to find. Before diving into operations, pinpointing the variable gives us a clear target for our mathematical explorations. In an exercise, the variable is typically right beside a number or operation, distinguishable from known values and constants.

Substitution Method

The substitution method is an effective technique to solve equations once you've identified the variable. In this context, substitution involves replacing the variable with a given value. Let's consider the equation \(3x + 4 = 19\) and the proposition that \(x = 5\). By substituting \(5\) into \(x\), our equation transforms: \(3 \times 5 + 4 = 19\). Why is substitution useful?

• It simplifies equations, allowing us to focus on numeric operations.
• It tests whether a proposed value satisfies the equation.

Substitution acts like replacing a symbol with a concrete value, bridging the abstract and tangible in algebra.

Multiplication and Addition Operations

Once the substitution is complete, we move on to multiplication and addition to simplify the equation. After substituting \(x = 5\), the equation becomes \(3 \times 5 + 4\).

• Multiplication: Perform the multiplication first. Here, \(3 \times 5 = 15\).
• Addition: Add the result of the multiplication to the constant value in the equation. That gives us \(15 + 4 = 19\).

These basic arithmetic operations are straightforward yet crucial. They finalize the check for whether the initial substituted value solves the equation. Completing these steps shows whether the left and right sides of the equation balance, validating the given solution.